Abstract

Our work describes a technique for testing the centricity of optical systems by using the point spread function. It is shown that a specific position of an axial object point can be found for every optical element, where the spherical aberration is either zero or minimal. If we image such a point with an optical element, then its point spread function will be almost identical to the point spread function of the diffraction-limited optical system. This consequence can be used for testing the centricity of precisely fabricated optical elements, because we can simply detect asymmetry of the point spread function, which is caused by the decentricity of the tested optical element. One can also use this method for testing optical elements in connection with a cementing process. Moreover, a simple formula is also derived for calculation of the coefficient of third-order coma, which is caused by the decentricity of the optical surface due to a tilt of the surface with respect to the optical axis, and a simple method for detecting the asymmetry of the point spread function is proposed.

© 2008 Optical Society of America

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References

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  1. A. Maréchal, Imagerie Géométrique Aberrations (Revue d'Optique, 1952).
  2. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, 1970).
  3. G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).
  4. M. Herzberger, Modern Geometrical Optics (Interscience, 1958).
  5. P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford U. Press, 1997).
  6. M. Born and E. Wolf, Principles of Optics (Oxford U. Press, 1964).
  7. H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).
  8. A. Cox, A System of Optical Design (Focal, 1964).
  9. H. Haferkorn, Bewertung optisher systeme (VEB Deutscher Verlag der Wissenschaften, 1986).
  10. W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).
  11. A. Mikš, Applied Optics (Czech Technical University Press, 2000).
  12. A. N. Bardin, Optical Glass Technology (Higher Schools, 1963).
  13. D. F. Horne, Optical Production Technology (Institute of Physics, 1982).
  14. H. H. Karow, Fabrication Methods for Precision Optics(Wiley, 1993).
  15. M. A. Okatov, Handbook of Optical Technology (Politekhnika, 2004).
  16. H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
    [CrossRef]
  17. M. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533-537 (1970).
    [CrossRef] [PubMed]
  18. B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A , 14, 2837-2849 (1997).
    [CrossRef]
  19. L. I. Epstein, “The aberrations of slightly decentered optical systems,” J. Opt. Soc. Am. 39, 847-853 (1949).
    [CrossRef]
  20. H. A. Buchdahl, “Perturbations of the point characteristic,” J. Opt. Soc. Am. A 7, 2260-2263 (1990).
    [CrossRef]
  21. G. Wooters, “Lens centering in microscope objectives,” J. Opt. Soc. Am. 40, 521-523 (1950).
    [CrossRef]
  22. P. L. Ruben, “Aberrations arising from decentrations and tilts,” J. Opt. Soc. Am. 54, 45-52 (1964).
    [CrossRef]
  23. R. Gelles, “Off-center aberrations in nonaligned systems,” J. Opt. Soc. Am. 68, 1250-1254 (1978).
    [CrossRef]
  24. A. Mikš, J. Novák, and P. Novák, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik (Jena) 118, 537-543 (2007).
    [CrossRef]
  25. D. Malacara, Optical Shop Testing (Wiley, 2007).
    [CrossRef]
  26. D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
    [CrossRef]
  27. A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A 19, 1867-1871 (2002).
    [CrossRef]

2007 (2)

A. Mikš, J. Novák, and P. Novák, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik (Jena) 118, 537-543 (2007).
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley, 2007).
[CrossRef]

2005 (1)

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

2004 (1)

M. A. Okatov, Handbook of Optical Technology (Politekhnika, 2004).

2002 (1)

2000 (1)

A. Mikš, Applied Optics (Czech Technical University Press, 2000).

1997 (2)

B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A , 14, 2837-2849 (1997).
[CrossRef]

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford U. Press, 1997).

1993 (1)

H. H. Karow, Fabrication Methods for Precision Optics(Wiley, 1993).

1990 (1)

1986 (1)

H. Haferkorn, Bewertung optisher systeme (VEB Deutscher Verlag der Wissenschaften, 1986).

1984 (1)

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

1982 (1)

D. F. Horne, Optical Production Technology (Institute of Physics, 1982).

1978 (1)

1974 (1)

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

1970 (2)

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, 1970).

M. Rimmer, “Analysis of perturbed lens systems,” Appl. Opt. 9, 533-537 (1970).
[CrossRef] [PubMed]

1966 (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

1964 (3)

P. L. Ruben, “Aberrations arising from decentrations and tilts,” J. Opt. Soc. Am. 54, 45-52 (1964).
[CrossRef]

A. Cox, A System of Optical Design (Focal, 1964).

M. Born and E. Wolf, Principles of Optics (Oxford U. Press, 1964).

1963 (1)

A. N. Bardin, Optical Glass Technology (Higher Schools, 1963).

1958 (1)

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

1952 (1)

A. Maréchal, Imagerie Géométrique Aberrations (Revue d'Optique, 1952).

1950 (2)

1949 (1)

Bardin, A. N.

A. N. Bardin, Optical Glass Technology (Higher Schools, 1963).

Born, M.

M. Born and E. Wolf, Principles of Optics (Oxford U. Press, 1964).

Buchdahl, H. A.

H. A. Buchdahl, “Perturbations of the point characteristic,” J. Opt. Soc. Am. A 7, 2260-2263 (1990).
[CrossRef]

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, 1970).

Cox, A.

A. Cox, A System of Optical Design (Focal, 1964).

Epstein, L. I.

Gelles, R.

Haferkorn, H.

H. Haferkorn, Bewertung optisher systeme (VEB Deutscher Verlag der Wissenschaften, 1986).

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

Hopkins, H. H.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).

Horne, D. F.

D. F. Horne, Optical Production Technology (Institute of Physics, 1982).

Karow, H. H.

H. H. Karow, Fabrication Methods for Precision Optics(Wiley, 1993).

Macdonald, J.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford U. Press, 1997).

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, 2007).
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

Maréchal, A.

A. Maréchal, Imagerie Géométrique Aberrations (Revue d'Optique, 1952).

Mikš, A.

A. Mikš, J. Novák, and P. Novák, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik (Jena) 118, 537-543 (2007).
[CrossRef]

A. Mikš, “Modification of the formulas for third-order aberration coefficients,” J. Opt. Soc. Am. A 19, 1867-1871 (2002).
[CrossRef]

A. Mikš, Applied Optics (Czech Technical University Press, 2000).

Mouroulis, P.

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford U. Press, 1997).

Novák, J.

A. Mikš, J. Novák, and P. Novák, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik (Jena) 118, 537-543 (2007).
[CrossRef]

Novák, P.

A. Mikš, J. Novák, and P. Novák, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik (Jena) 118, 537-543 (2007).
[CrossRef]

Okatov, M. A.

M. A. Okatov, Handbook of Optical Technology (Politekhnika, 2004).

Rimmer, M.

Ruben, P. L.

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

Slyusarev, G. G.

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

Stone, B. D.

Tiziani, H. J.

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

Welford, W. T.

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Oxford U. Press, 1964).

Wooters, G.

Appl. Opt. (1)

Br. J. Appl. Phys. (1)

H. H. Hopkins and H. J. Tiziani, “A theoretical and experimental study of lens centring errors and their influence on optical image quality,” Br. J. Appl. Phys. 17, 33-54 (1966).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

Optik (Jena) (1)

A. Mikš, J. Novák, and P. Novák, “Calculation of point-spread function for optical systems with finite value of numerical aperture,” Optik (Jena) 118, 537-543 (2007).
[CrossRef]

Other (17)

D. Malacara, Optical Shop Testing (Wiley, 2007).
[CrossRef]

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Taylor & Francis, 2005).
[CrossRef]

A. Maréchal, Imagerie Géométrique Aberrations (Revue d'Optique, 1952).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, 1970).

G. G. Slyusarev, Aberration and Optical Design Theory (Adam Hilger, 1984).

M. Herzberger, Modern Geometrical Optics (Interscience, 1958).

P. Mouroulis and J. Macdonald, Geometrical Optics and Optical Design (Oxford U. Press, 1997).

M. Born and E. Wolf, Principles of Optics (Oxford U. Press, 1964).

H. H. Hopkins, Wave Theory of Aberrations (Oxford U. Press, 1950).

A. Cox, A System of Optical Design (Focal, 1964).

H. Haferkorn, Bewertung optisher systeme (VEB Deutscher Verlag der Wissenschaften, 1986).

W. T. Welford, Aberrations of the Symmetrical Optical Systems (Academic, 1974).

A. Mikš, Applied Optics (Czech Technical University Press, 2000).

A. N. Bardin, Optical Glass Technology (Higher Schools, 1963).

D. F. Horne, Optical Production Technology (Institute of Physics, 1982).

H. H. Karow, Fabrication Methods for Precision Optics(Wiley, 1993).

M. A. Okatov, Handbook of Optical Technology (Politekhnika, 2004).

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Figures (3)

Fig. 1
Fig. 1

Calculation of PSF: W 31 = 0.5 λ , (a)  W 11 = 0 , (b)  W 11 = 2 / 3 W 31 .

Fig. 2
Fig. 2

Dependence of correlation coefficient R on coma aberration coefficient W 31 ( W 11 = 2 / 3 W 31 ).

Fig. 3
Fig. 3

PSF at the optimal image point ( α = 1 , W 40 = 0.05 λ , W 31 0 = 1.29 λ ).

Tables (3)

Tables Icon

Table 1 Parameters of the Optical System

Tables Icon

Table 2 Aberration of Optical System

Tables Icon

Table 3 Wave Aberration with Respect to Reference Sphere

Equations (56)

Equations on this page are rendered with MathJax. Learn more.

S I / u 3 = a 3 m 3 + a 2 m 2 + a 1 m + a 0 ,
a 3 = S I 0 + 4 S II 0 2 ( 2 + p ) ,
a 2 = 3 S I 0 8 S II 0 + 2 ( 2 + p ) + 1 ,
a 1 = 3 S I 0 + 4 S II 0 1 , a 0 = S I 0 ,
δ s = h 2 2 f S I 0 , δ f = h 2 2 f ( S I 0 S II 0 ) ,
S I 0 = 2 f h 2 δ s , S II 0 = 2 f h 2 ( δ f δ s ) .
a 3 m 3 + a 2 m 2 + a 1 m + a 0 = 0 .
( m 1 ) ( b 2 m 2 + b 1 m + b 0 ) = 0 ,
b 2 = a 3 = S I 0 + 4 S II 0 2 ( 2 + p ) ,
b 1 = b 2 + a 2 = 2 S I 0 4 S II 0 + 1 ,
b 0 = b 1 + a 1 = S I 0 .
b 2 m 2 + b 1 m + b 0 = 0 .
D = b 1 2 4 b 2 b 0 0 .
D = 16 S II 0 2 8 S II 0 4 S I 0 ( 3 + 2 p ) + 1 .
D = 16 S II 0 2 8 S II 0 4 S I 0 ( 3 + 2 p ) + 1 = 0.
S II 0 = 0.25 ± 0.5 ( 3 + 2 p ) S I 0 .
S II 0 0.25 0.5 ( 3 + 2 p ) S I 0 , S II 0 0.25 + 0.5 ( 3 + 2 p ) S I 0 .
S I m = 3 a 3 m 2 + 2 a 2 m + a 1 = 0 .
D e = 4 ( a 2 2 3 a 3 a 1 ) 0 .
1 s 1 s = 1 f , m = s s ,
s = f ( 1 m ) , s = s / m .
A = D 0 2 s = D 0 m 2 f ( 1 m ) , A = A m = D 0 2 f ( 1 m ) ,
δ W i R i ( n i cos I i n i cos I i n i + n i ) α i cos ( φ i θ i ) ,
δ W i Y i 2 ( 1 n i 1 n i ) n i 2 sin 2 I i α i ,
1 cos φ 1 2 sin 2 φ .
n sin I = n ( r s r ) sin U n Y ( 1 r 1 s ) = Y Q ,
δ W i 1 2 ( 1 n i 1 n i ) Q i 2 Y i 3 α i = [ 1 2 ( 1 n i 1 n i ) H i 3 Q i 2 α i ] ( Y i H i ) 3 = ( W 31 ) i y 3 ,
δ W i 0 1 2 ( 1 n i 1 n i ) Q i 2 Y i 3 α i + ( n i n i ) Y i α i = δ W i + ( n i n i ) Y i α i .
U ( P ) = i λ s U ( M ) e i k r r cos ( n , r ) d S ,
p = ( x x 0 ) / R , q = ( y y 0 ) / R ,
s = n x P x 0 λ 0 , t = n y P y 0 λ 0 ,
F ( p , q ) = U ( p , q ) 1 p 2 q 2 exp ( i k 0 W ) ,
U ( s , t ) = C s F ( p , q ) exp [ 2 π i ( p s + q t ) ] d p d q ,
U ( s , t ) = C s exp ( i k 0 W ) exp [ 2 π i ( p s + q t ) ] d p d q .
S.D. = 1 k 2 ( W 2 ¯ W ¯ 2 ) = 1 k 2 E 0 ,
W ¯ = 1 S s W d S , W 2 ¯ = 1 S s W 2 d S .
E 0 = W 2 ¯ W ¯ 2 λ 2 197 ,
W = W 11 r cos φ + W 20 r 2 + W 40 r 4 + W 31 r 3 cos φ ,
S.D. = 1 k 2 ( 1 12 W 20 2 + 1 6 W 20 W 40 + 4 45 W 40 2 + 1 4 W 11 2 + 1 3 W 11 W 31 + 1 8 W 31 2 ) .
W 20 = W 40 , W 11 = ( 2 / 3 ) W 31 .
W optim = W 40 r 2 ( r 2 1 ) + W 31 r ( r 2 2 / 3 ) cos φ .
( S.D. ) max = 1 k 2 ( W 40 2 180 + W 31 2 72 ) = 1 k 2 180 ( W 40 2 + 2.5 W 31 2 ) .
R = 0.01577 ( W 31 λ ) 6 0.01254 ( W 31 λ ) 4 0.04396 ( W 31 λ ) 2 + 1 .
( W 31 / λ ) 2 = 1.82378 [ 1 ( S.D. ) max ] .
R = 0.02876 [ 1 ( S.D. ) max ] 3 0.02287 [ 1 ( S.D. ) max ] 2 0.08017 [ 1 ( S.D. ) max ] + 1 .
S I 0 = 2 f h 2 δ s , S II 0 = 2 f h 2 ( δ f δ s ) ,
0.138 m 2 1.044 m 3.394 = 0 .
s 1 = 22.975 + s H = 21.105 mm ,
s 1 = 230.20 + s H = 231.05 mm,
s 2 = 35.920 + s H = 34.095 mm,
s 2 = 88.167 + s H = 87.320 mm ,
A 1 = 0.087 , A 1 = 0.0087 ,
A 2 = 0.056 , A 2 = 0.0226 .
W 31 0 = i | W i 0 α i | = 2.53 α ( i = 1 , 2 , 3 ) .
W 0 = W 40 r 4 + W 31 0 r 3 cos φ .
W optim = W 40 r 2 ( r 2 1 ) + W 31 0 r ( r 2 2 / 3 ) cos φ .

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